Information has been transmitted over optical fibers since the late 1970s. Discussions in this field are disclosed in “Optical Communication Systems” by John Gowar (Prentice Hall, 2nd ed., 1993) and “Fiber-optic communication systems” by Govind P. Agrawal (Wiley, 2nd ed., 1997), which are herein incorporated by reference. The information is usually in the form of binary digital signals, i.e. logical “1”s and “0”s, but fiber optics is also used to transport analog signals. The remainder of this document will refer only to the applications with digital signals. Every transmission system has a transmitter, which emits light modulated with information into the fiber, and a receiver at the far end which detects the light and recovers the information. A typical transmission system might have several spans of optical fiber with erbium doped fiber amplifiers (EDFAs) between spans. The EDFAs amplify the optical signal to overcome the loss of the fiber spans. The total transmission distance through optical fiber experienced by an optical signal may be several thousand kilometers.
The simplest way of imposing information onto the optical carrier at the transmitter is by modulation of the amplitude (or power or intensity) of the light. For binary digital signals this corresponds to on-off modulation. The receiver then comprises a simple photodetector, employing direct detection. The photocurrent generated by the photodetector is a replica of the power falling on the photodetector. Subsequent electronic circuits amplify and process the photocurrent electrical signal to determine the information content of the received optical signal. Alternatively it is possible to modulate information onto the electric field of the optical carrier. There are several advantages to imposing information by modulating the electric field, but it is not yet in widespread use because the receiver is more complex. A simple direct detection receiver cannot be used, because it responds to the power (the absolute value squared of the electric field) and not to the electric field of the optical signal. Thus, any information in the phase of the optical signal is lost. A coherent detection receiver may be used, as this type of receiver does respond to the optical signal's electric field. In a coherent receiver, the incoming optical signal is mixed with continuous wave light from a local oscillator of the same wavelength, and then detected. The photocurrent in the photodetector includes a term which is the beat product of the optical signal and local oscillator, and which depends on the optical signal's electric field. Typically further processing is needed to obtain the electric field from the beat product. U.S. Patent Application 2004/0114939, herein incorporated by reference, discloses a phase diverse coherent receiver configuration using digital signal processing (DSP) to calculate the electric field. Values of the real and imaginary parts (the inphase and quadrature components) of the complex electric field are then available within the digital processor for further processing.
The optical signal may be distorted by propagation through the optical fiber. There are several distinct propagation effects that can occur, as described in “Nonlinear fiber optics” by Govind P. Agrawal (Academic Press, 2nd ed., 1995). Chromatic dispersion (CD) is the propagation effect most often encountered. A 10 Gb/s on-off modulated optical signal is substantially distorted by CD after propagation through about 100 km of non-dispersion shifted fiber (NDSF), so it is necessary to compensate for chromatic dispersion in some way in order to transmit over longer distances than 100 km. The usual way to compensate for CD is via dispersion compensation fiber (DCF), an optical component placed in line with the transmission fiber. DCF has the positive feature that it compensates exactly for chromatic dispersion, but it has disadvantages that it is expensive, it is physically large in size, it has substantial optical loss, and the amount of CD being compensated is fixed. There are ways to compensate for chromatic dispersion in the electronics of the receiver after photodetection. For example, “Adaptive Electronic Feed-Forward Equaliser and Decision Feedback Equaliser for the Mitigation of Chromatic Dispersion and PMD in 43 Gbit/s Optical Transmission Systems” by B. Franz et al. (ECOC 2006 conference, Cannes, France, paper We1.5.1, September 2006) describes an electronic domain CD compensator using analog signal processing, and “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation” by A. Farbert et al. (ECOC 2004 conference, Stockholm, Sweden, paper Th4.1.5, September 2004) describes one using digital signal processing. Compensation in the electrical domain is expected to cost less than using optical components because of the nature of mass production of electronics. However electrical compensation of CD following direct detection can only compensate for a small amount of chromatic dispersion, equivalent to perhaps 100 km of NDSF at 10 Gb/s, because direct detection discards the phase of the optical signal. For this reason on long fiber optic transmission systems most of the CD compensation has been done via DCF, with electrical domain compensation simply trimming the amount of compensation.
Recently two new methods of electrical domain CD compensation have been proposed which are able in principle to compensate for unlimited transmission distances. Both methods use digital signal processing and effectively operate on a discrete-time representation of the electric field of the optical signal. The first method precompensates for chromatic dispersion, and is disclosed in U.S. Pat. No. 7,023,601. An optical signal is transmitted which may not resemble the information content, and after propagation through optical fiber the chromatic dispersion of the fiber transforms the optical signal into the desired form, which does resemble the information content. A pair of Mach-Zehnder optical modulators in the transmitter allow the inphase and quadrature parts (the real and imaginary parts) of the electric field to be modulated independently. Each of the two Mach-Zehnder modulators is driven by an electrical signal set by a digital-to-analog (D/A) converter, which in turn, is controlled by a digital signal processor. The DSP calculates the electric field of the precompensated optical signal such that after propagating through the known chromatic dispersion of the fiber optic transmission system the correct optical signal arrives at the receiver. The receiver may be a direct detection receiver, given that the precompensated optical signal is calculated to become an on-off modulated signal after propagation through the optical fiber. Alternatively, the receiver may be of a more advanced design, such as a coherent receiver, and the optical signal arriving at the receiver may then be a phase modulated signal.
The second method of CD compensation that operates on the electric field of the optical signal is with a phase diverse coherent receiver, as described in U.S. Patent Application Number 2004/0114939. A conventional optical signal is transmitted, such as an on-off modulated signal or a phase modulated signal. The optical signal becomes distorted by the chromatic dispersion of the fiber optic transmission system. The coherent receiver uses DSP to calculate the electric field of the incoming optical signal, and these electric field values can then be acted upon to compensate for the effect of the chromatic dispersion.
The present invention is related to the calculation performed within the digital signal processor to compensate for chromatic dispersion. Although the two methods differ in that one precompensates for CD at the transmitter, while the other postcompensates at the receiver, the calculation is very similar. The digital signal processor takes the given electric field, either the undistorted signal in the case of precompensation or the distorted signal in the case of postcompensation, and calculates the impact of an element of chromatic dispersion having the same magnitude and the opposite sign to the actual chromatic dispersion of the fiber optic transmission system. The calculation is typically done by a finite impulse response (FIR) filter, also known as a transversal filter. FIG. 1 illustrates a structure of an FIR filter. It comprises several delay stages 102 and multiply stages 104, and the multiplications results are summed 106. Each delay r corresponds to one sample of the digitized representation, which is typically half a digital symbol period. The FIR filter implements the following equation:
                              y          ⁡                      (            n            )                          =                              ∑                          k              =              0                                      N              -              1                                ⁢                                    x              ⁡                              (                                  n                  -                  k                                )                                      ⁢                          h              ⁡                              (                k                )                                                                        (        1        )            
where x(n) are the input electric field values, y(n) are the output values, and h(k) are the tap weights. All three variables are complex numbers, and the multiplication appearing on the right hand side is complex multiplication. n is the sample number, incrementing at typically two times per digital symbol, and N refers to the number of filter taps. The FIR filter is implemented as digital logic gates in an integrated circuit, and so it does not follow that the structure of FIG. 1 will appear in the integrated circuit. The tap weight coefficients h(k) may be determined from the inverse Fourier transform of the transfer function of the required amount of chromatic dispersion. For example, FIG. 2 is a graph illustrating the real and imaginary parts of the tap weights to compensate for 2000 km NDSF at a symbol rate of 10 Gbaud. Each point in time in the graph of FIG. 2 corresponds to one tap weight. 140 delay and multiply stages are needed to accurately compensate for chromatic dispersion in this case. In fact, the set of tap weight coefficients associated with chromatic dispersion is always symmetric about the center, so it is possible to use each multiplication result twice. Taking that saving into account, there are still 70 multiplications to be executed for each symbol period. This number is so large, that in practice, most of the computations performed by the digital signal processor are for chromatic dispersion compensation, and the amount of computations are on the edge of what is possible using today's integrated circuit technology. In addition, the power dissipation of the integrated circuit is proportional to the amount of computations, so it is beneficial for that reason to reduce the amount of computations.
The variables x(n), y(n) and h(k) are in general continuous quantities, and they can take on a smooth range of values. To compute equation 1, these variables are approximated by a discrete set of values, which is a finite length binary digital representation. Additionally, the D/A converter used with precompensation, and the A/D converter for the postcompensation case, work on discrete values of a certain number of binary digits. When a small number of binary digits is used to represent a variable, it makes the CD compensation less accurate, but it saves on integrated circuit resources.
While the CD compensation calculation for precompensation is similar to postcompensation, there are important differences between the amounts of computations needed to implement the two of them. With precompensation, the variable x(n) in equation 1 refers to the electric field of the optical signal containing the specified information, and y(n) refers to the desired transmit signal electric field to be sent to the D/A converters. Depending on the modulation format chosen, x(n) takes on only values from a small set. For example, if on-off modulation is used x(n) can take on two possible values from the set {0,1} at the symbol centers. Alternatively with quadrature phase shift keying (QPSK) modulation format x(n) takes on one of four possible values at the symbol centers, {−1,1,−i,i}, where i refers to the imaginary number √{square root over (−1)}. In both these modulation format examples, any of the allowable values of x(n) is written as a short number in a binary digital representation. This means that the product terms x(n−k) h(k) of equation 1 may be evaluated with a small number of logic gates. In contrast, when postcompensation is implemented in conjunction with coherent detection, the variable x(n) refers to the electric field of the incoming optical signal, and y(n) refers to the signal after chromatic dispersion compensation. x(n) may take on a wide range of values in this case, typically limited by the resolution of the D/A converter. It requires several binary digits to adequately represent x(n), and there is no economy in the number of logic gates to calculate the terms x(n−k) h(k). For this reason, precompensation typically requires fewer computation resources than postcompensation for an equivalent quality of CD compensation. However there are other benefits to using a coherent receiver, and it is desirable to implement CD compensation in a coherent receiver using the same amount of computations or fewer than precompensation.
An alternative to the FIR filter is to use the discrete Fourier transform (DFT) for CD compensation. The DFT is calculated for a block of contiguous sample values, where the size of the block is typically several times larger than the extent of the CD impulse response. The discrete Fourier transform may be calculated using one of the well-established multistage fast Fourier transform algorithms which are described in “Understanding Digital Signal Processing” by Richard G. Lyons (Prentice Hall, 1996), herein incorporated by reference. The fast Fourier transform algorithm calculates the Fourier transform of N points in log2N stages, each stage involving the multiplication and addition of pairs of values. The output values of the DFT (the spectrum of the input) are multiplied by the phase factors associated with the chromatic dispersion transfer function. Then the inverse discrete Fourier transform is calculated by a similar algorithm to the forward DFT, to produce a discrete-time sequence which represents the CD compensated optical signal as a function of time. The DFT may use fewer arithmetic operations compared to the FIR filter. For example, in the case of compensation for 2000 km NDSF at 10 Gbaud with a 512-point DFT window, the number of multiplications per symbol is 44. However, more buffering of data values is needed since the DFT window is larger than the CD impulse response (the FIR filter width), so the reduction in number of multiplications may be offset by the extra latches and communication resources needed in the integrated circuit.
A second alternative way of performing CD compensation to the FIR filter has been proposed and modeled. A solution using an infinite impulse response (IIR) filter design is described in “Chromatic Dispersion Compensation Using Digital IIR Filtering With Coherent Detection” by G. Goldfarb & G. Li (IEEE Phot. Tech. Lett., vol. 19, no. 13, p. 969-971, 2007). It is well known that the transfer function of an FIR filter having many taps may often be implemented more compactly by an IIR filter. Goldfarb & Li obtained a reduction by a factor of 2.5 in the amount of computations compared to the FIR filter. However, the digital signal processor in an actual implementation is likely to be organized in a parallel architecture, and there is a difficulty implementing an IIR filter in a parallel digital processor. The IIR filter inherently uses feedback from previous results, such as y(n−1), to calculate result y(n). In contrast, the FIR filter of equation 1 does not have any terms in y(n−1) on the right hand side. In a parallel architecture digital processor, the result y(n−1) may not be available at the time of calculating y(n), so the IIR algorithm cannot be implemented. This issue is discussed in U.S. Patent Application Number 2006/0245766, herein incorporated by reference. There are ways to resolve the problem, such as recasting the algorithm using a look-ahead computation, but the solution requires more computations. It is possible that the IIR filter for CD compensation described by Goldfarb & Li would reduce the amount of computations by significantly less than the factor of 2.5, if it were implemented in a parallel digital signal processor. It is desirable to find a way to adapt the IIR filter solution so that it can be implemented in a parallel digital signal processor without requiring excess computations.
Thus, there is a need for an algorithm to compensate for chromatic dispersion in a digital signal processor which uses a smaller amount of computations than a direct implementation of an FIR filter. It is preferable that such an algorithm does not require buffering of a larger number of sample values than the FIR filter algorithm. Also there is a need for an IIR filter algorithm that uses a small amount of computations when implemented on a parallel digital signal processor.